
For the CMB, we consider the shift parameters \((R, l_a)\)  and \(\Omega _{b0}h_0^2\) , with \(H_0=100h_0\) , based on Planck final release[1]} as derived by [2]}.

For the CMB, we also consider its temperature that constrains \(\Omega _{r0}\)  and avoids its degeneracy with the \(\alpha \)  parameter. \(\Omega _{r0}\)  can be separated in two components, one for the photons \(\Omega _{\gamma 0}\)  and one for the relativistic neutrinos \(\Omega _{\nu 0}\) . The photons density \(\Omega _{\gamma 0}h_0^2\)  is precisely measured by the CMB temperature as \(\Omega _{\gamma 0}h_0^2=2.47282\times 10^{-5}\) . \(\Omega _{\nu 0}\)  is related to \(\Omega _{\gamma 0}\)  by(see [3]} for instance) \(\Omega _{\nu 0}=0.2271\Omega _{\gamma 0} N_{eff}\)  where \(N_{eff}\)  is the effective number of neutrino species. Then, the values \(h_0=0.674\pm 0.005\)  and \(N_{eff}=2.99\pm 0.17\)  by Planck[1]} lead to the \(1\sigma \)  constraint \(\Omega _{r0}=(9.13\pm 0.25)\times 10^{-5}\)  that we introduce as a prior in the \(\chi ^2\)  defined below.

For the supernovae, we consider the 1048 supernovae from the last Pantheon data[5]}.

For the BAO measurements, we consider the Ly\(\alpha \)  forest from BOSS DR11[6]}, BOSS DR12[7]}, WiggleZ[8]}, 6dFGS[9]} and DES Year 1[10]}. This corresponds to 10 measurements.

For the Hubble function, we consider 36 data points from [11]}, [12]}, [13]}, [14]} and [15]}.

